# Solve. displaystyleintfrac{{{78}{x}}}{{{25}-{12}{x}+{4}{x}^{2}}}

Question
Integrals
Solve. $$\displaystyle\int\frac{{{78}{x}}}{{{25}-{12}{x}+{4}{x}^{2}}}$$

2020-12-16
The given integration is
$$\displaystyle\int\frac{{{78}{x}}}{{{25}-{12}{x}+{4}{x}^{2}}}$$
A simple calculation the integration becomes
$$\displaystyle\int\frac{{{78}{x}}}{{{25}-{12}{x}+{4}{x}^{2}}}={7}\int\frac{{{\left.{d}{x}\right.}}}{{{25}-{12}{x}+{4}{x}^{2}}}$$
$$\displaystyle={7}\int\frac{{{\left.{d}{x}\right.}}}{{{4}{x}^{2}-{12}{x}+{25}}}$$
$$\displaystyle={7}\int\frac{{{\left.{d}{x}\right.}}}{{{4}{\left({x}^{2}-{3}{x}+\frac{25}{{4}}\right)}}}$$
$$\displaystyle=\frac{7}{{4}}\int\frac{{{\left.{d}{x}\right.}}}{{{x}^{2}-{3}{x}+\frac{25}{{4}}}}$$
$$\displaystyle=\frac{7}{{4}}\int\frac{{{\left.{d}{x}\right.}}}{{{x}^{2}-{3}{x}+\frac{25}{{4}}}}$$
$$\displaystyle=\frac{7}{{4}}\int\frac{{{\left.{d}{x}\right.}}}{{{x}^{2}-{2}\cdot{x}\cdot\frac{3}{{2}}+\frac{9}{{4}}+\frac{25}{{4}}-\frac{9}{{4}}}}$$
$$\displaystyle=\frac{7}{{4}}\int\frac{{{\left.{d}{x}\right.}}}{{{\left({x}-\frac{3}{{2}}\right)}^{2}+\frac{16}{{4}}}}$$
$$\displaystyle=\frac{7}{{4}}\int\frac{{{\left.{d}{x}\right.}}}{{{\left({x}-\frac{3}{{2}}\right)}^{2}+{\left({2}\right)}^{2}}}$$
$$\displaystyle=\frac{7}{{4}}\times\frac{1}{{2}}{{\tan}^{ -{{1}}}{\left(\frac{{{x}-\frac{3}{{2}}}}{{2}}\right)}}+{C}$$
$$\displaystyle{\left[\text{By using the formula}\ \int\frac{{{\left.{d}{x}\right.}}}{{x}^{2}}+{a}^{2}=\frac{1}{{a}}{{\tan}^{ -{{1}}}{\left(\frac{x}{{a}}\right)}}+{C}\right]}$$
$$\displaystyle=\frac{7}{{8}}{{\tan}^{ -{{1}}}{\left(\frac{{{2}{x}-{3}}}{{4}}\right)}}+{C}$$
Where C is the constant of integration.
Thus, the value of the integration is
$$\displaystyle\int\frac{{{7}{\left.{d}{x}\right.}}}{{{25}-{12}{x}+{4}{x}^{2}}}=\frac{7}{{8}}{{\tan}^{ -{{1}}}{\left(\frac{{{2}{x}-{3}}}{{4}}\right)}}+{C}$$

### Relevant Questions

Solve. $$\displaystyle\int\frac{{\sqrt{{x}}}}{{{1}+{\sqrt[{{3}}]{{x}}}}}{\left.{d}{x}\right.}=?$$
Evaluate the integral $$\displaystyle\int\frac{x}{{2}}{\left.{d}{x}\right.}$$
Give the correct answer and solve the given equation Evaluate $$\displaystyle\int{x}^{3}{\left({\sqrt[{3}]{{{1}-{x}^{2}}}}\right)}{\left.{d}{x}\right.}$$
Give the correct answer and solve the given equation $$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}=\frac{{{y}^{2}-{1}}}{{{x}^{2}-{1}}},{y}{\left({2}\right)}={2}$$
Give the correct answer and solve the given equation
Let $$\displaystyle{p}{\left({x}\right)}={2}+{x}{\quad\text{and}\quad}{q}{\left({x}\right)}={x}$$. Using the inner product $$\langle\ p,\ q\rangle=\int_{-1}^{1}pqdx$$ find all polynomials $$\displaystyle{r}{\left({x}\right)}={a}+{b}{x}\in{P}{1}{\left({R}\right)}{P}$$
(R) such that {p(x), q(x), r(x)} is an orthogonal set.
Give the correct answer and solve the given equation $$\displaystyle{\left({x}-{y}\right)}{\left.{d}{x}\right.}+{\left({3}{x}+{y}\right)}{\left.{d}{y}\right.}={0},\text{when}\ {x}={3},{y}=-{2}$$
Evaluate the triple integral $$\displaystyle\int\int\int_{{E}}{3}{y}{d}{V}$$,where
$$\displaystyle{E}={\left\lbrace{\left({x},{y},{z}\right)}{\mid}{0}\le{x}\le{2},{0}\le{y}\le\sqrt{{{4}-{x}^{{2}}}},{0}\le{z}\le{x}\right\rbrace}$$
1. Find each of the requested values for a population with a mean of $$? = 40$$, and a standard deviation of $$? = 8$$ A. What is the z-score corresponding to $$X = 52?$$ B. What is the X value corresponding to $$z = - 0.50?$$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $$M=42$$ for a sample of $$n = 4$$ scores? E. What is the z-scores corresponding to a sample mean of $$M= 42$$ for a sample of $$n = 6$$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $$a. -2.00 b. 1.25 c. 3.50 d. -0.34$$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $$\mu = 78$$ and $$\sigma = 12$$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $$82, 74, 62, 68, 79, 94, 90, 81, 80$$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $$12 (\mu = 42, \sigma = 12)$$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is$44.50 from tips. Test for a difference between this value and the population mean at the $$\alpha = 0.05$$ level of significance.
Solve. $$int (4ax^3+3bx^2+2cx)dx$$
$$\displaystyle{\left({x}+{y}\right)}{\left.{d}{x}\right.}+{\left({x}-{y}\right)}{\left.{d}{y}\right.}={0}$$